Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2104 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#11319 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #401 |
| commutator | C24 | C22×C4×C7 |
| Frattini, cosocle | C23×C4 | C23×C7 |
| socle | C2×C7 | #1443 |
| prime radical | C24×C7 | C22×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C23×C4×C7 ◁ G | |
532 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,12,17,13,16,25,32)(10,14,20,28,19,27,24,33)(11,21,23,36,22,15,29,31)(18,26,35,38,34,30,37,39);(8,10)(9,14,16,27)(11,18,22,34)(12,24)(13,19)(15,30)(17,33,32,28)(20,25)(21,26)(23,37,29,35)(31,38)(36,39);(8,11)(9,15,16,21)(10,18,19,34)(12,23)(13,22)(14,26)(17,31,32,36)(20,35,24,37)(25,29)(27,30)(28,38)(33,39)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 4307);